Hep-Th/0604077

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Inspired by the work on spectrum generating Lie algebras by physicists [1] Joseph introduced the concept of minimal unitary realizations of Lie algebras. It is basically defined as a realization that exponentiates to a unitary representation of the corresponding noncompact group on a Hilbert space of functions depending on the minimal number of coordinates. Joseph gave the minimal realizations of the complex forms of classical Lie algebras and of G2 in a Cartan-Weil basis [2, 3]. The existence of the minimal unitary representation of E8(8) within the framework of Langland’s classification was first proved by Vogan [4] . Later, the minimal unitary representations of all simply laced groups, were studied by Kazhdan and Savin[5], and Brylinski and Kostant [6, 7, 8, 9]. Gross and Wallach studied the minimal representations of quaternionic real forms of exceptional groups [10]. For a review and the references on earlier work on the subject in the mathematics literature prior to 2000 we refer the reader to the review lectures of Jian-Shu Li [11]. The idea that the theta series of E8(8) and its subgroups may describe the quantum supermembrane in various dimensions [12], led Pioline, Kazh- dan and Waldron [13] to reformulate the minimal unitary representations of simply laced groups[5]. In particular, they gave explicit realizations of the simple root (Chevalley) generators in terms of pseudo-differential operators for the simply laced exceptional groups, together with the spherical vectors necessary for the construction of modular forms. Motivated mainly by the idea that the spectra of toroidally compacti- fied M/superstring theories must fall into unitary representations of their U-duality groups and towards the goal of constructing these unitary representations Günaydin, Koepsell and Nicolai first studied the geometric realizations of U-duality groups of the corresponding supergravity theories [14]. In particular they gave geometric realizations of the U-duality groups of maximal supergravity in four and three dimensions as conformal and quasiconformal groups, respectively. The realization of the 3-dimensional U-duality group E8(8) of maximal supergravity given in [14] as a quasiconformal group that leaves invariant a generalized light-cone with respect to a quartic norm in 57 dimensions is the first known geometric realization of E8. An E7(7) covariant construction of the minimal unitary representation of E8(8) by quantization of its geometric realization as a quasi-conformal group [14] was then given in [16]. The minimal unitary realization of the 3 dimensional U-duality group E8(−24) of the exceptional supergravity [27] by quantization of its geometric realization as a quasiconformal group was subsequently given in [15]. By consistent truncation the quasiconformal re-

1 alizations of the other noncompact exceptional groups can be obtained from those of E8(8) and E8(−24). Apart from being the first known geometric realization of the exceptional group of type E8 the quasiconformal realization has some remarkable features. First, there exist different real forms of all simple groups that admit realizations as quasiconformal groups 3. There- fore, the quasiconformal realizations give a geometric meaning not only to the exceptional groups that appear in the last row of the famous Magic Square [28] but also extend to certain real forms of all simple groups. Another remarkable property of the quasiconformal realizations is the above mentioned fact that their quantization leads, in a direct and simple manner, to the minimal unitary representations of the corresponding noncompact groups [16, 15, 17]. Classification of the orbits of the actions of U-duality groups on the BPS black hole solutions in maximal supergravity and N = 2 Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions given in [30] suggested that four dimensional U-duality may act as a spectrum generating conformal symmetry in five dimensions [30, 14]. Furthermore, the work of [14] suggested that the 3 dimensional U duality group E8(8) of maximal supergravity must similarly act as a spectrum generating quasiconformal symmetry group in the charge space of BPS black hole solutions in four dimensions extended by an extra coordinate which was interpreted as black hole entropy. This extends naturally to 3-dimensional U-duality groups of N = 2 MESGTs acting as spectrum generating quasiconformal symmetry groups in four dimensions [31]. More recently it was conjectured that the indexed degeneracies of certain N = 8 and N = 4 BPS black holes are given by some automorphic forms related to the minimal unitary representations of the corresponding 3 dimensional U-duality groups [32]. Motivated by the above mentioned results and conjectures stationary and spherically symmetric solutions of N 2 supergravities with symmetric ≥ scalar manifolds were recently studied in [33]. By utilizing the equivalence of four dimensional attractor flow with the geodesic motion on the scalar manifold of the corresponding three dimensional theory the authors of [33] quantized the radial attractor flow, and argued that the three-dimensional U-duality groups must act as spectrum generating symmetry for BPS black hole degeneracies in 4 dimensions. They furthermore suggested that these degeneracies may be related to Fourier coefficients of certain modular forms of the 3-dimensional U-duality groups, in particular those associated with 3For SU(1, 1) = SL(2, R)= Sp(2, R) the quasiconformal realization reduces to conformal realization.

2 their minimal unitary representations. The quasiconformal realizations of noncompact groups represent natural extensions of generalized conformal realizations of some of their subgroups and were studied from a spacetime point of view in [17]. The authors of [17] studied in detail the quasiconformal groups of generalized spacetimes defined by Jordan algebras of degree three. The generic Jordan family of Euclidean Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra “dilatonic” coordinate, whose rotation, Lorentz and conformal groups are SO(d 1), SO(d 1, 1) SO(1, 1) − − × and SO(d, 2) SO(2, 1), respectively. The generalized spacetimes described × by simple Euclidean Jordan algebras of degree three correspond to extensions of Minkowskian spacetimes in the critical dimensions (d = 3, 4, 6, 10) by a dilatonic and extra (2, 4, 8, 16) commuting spinorial coordinates, respectively. Their rotation, Lorentz and conformal groups are those that occur in the first three rows of the Magic Square [28]. For the generic Jor- dan family the quasiconformal groups are SO(d + 2, 4). On the other hand, the quasiconformal groups of spacetimes defined by simple Euclidean Jordan algebras of degree are F4(4), E6(2), E7(−5) and E8(−24). The conformal subgroups of these quasiconformal groups are Sp(6, R),SU ∗(6), SO∗(12) and E7(−25), respectively. In this paper we give a unified construction of the minimal unitary representations of noncompact groups by quantization of their geometric realizations as quasiconformal groups and extend it to the construction of the minimal representations of noncompact supergroups. In section 2 we explain the connection between minimal unitary representations of noncompact groups G and their unique para-quaternionic symmetric spaces of the form G/H SL(2, R), which was used in [21] to give a classification and × minimal realizations of the real forms of infinite dimensional nonlinear quasi- superconformal Lie algebras that contain the Virasoro algebra as a subalgebra [20]. In section 3, using some of the results of [21] we give a unified construction of the minimal unitary representations of simple noncompact groups with H simple or Abelian. A degenerate limit of the unified construction leads to the minimal unitary representations of the symplectic groups Sp(2n, R), which is discussed in section 4. In sections 5, 6 ,7 and 8 we give the minimal unitary realizations of SO(p+2,q+2), SO∗(2n+4), SU(n+1,m+1) and SL(n + 2, R), respectively. In section 9 we extend our construction to the minimal realizations of noncompact supergroups and present the unified construction of the minimal representations of supergroups whose even subgroups are of the form H SL(2, R) with H simple.The construction of × the minimal unitary realizations of OSp(N 2, R) corresponds to a degener- |

3 ate limit of the uniﬁed construction and is discussed in section 10, where we also give the minimal realization of D(2, 1; α). Preliminary results of sections 3 and 9 appeared in [29].

2 Minimal Unitary Representations of Noncom- pact Groups and Para-Quaternionic Symmetric Spaces

The minimal dimensions for simple non-compact groups were determined by Joseph [2, 3]. For a particular noncompact group G the minimal dimension ℓ can be found by considering the 5-graded decomposition of its Lie algebra g, determined by a distinguished sl(2, R) subalgebra, of the form

g = g−2 g−1 g0 ∆ g+1 g+2 (2.1) ⊕ ⊕ ⊕ ⊕ ⊕ where g±2 are 1-dimensional subspaces each, and ∆ is the dilatation generator that determines the five grading. The generators belonging to the subspace g−2 ∆ g+2 form the sl(2 , R) subalgebra in question. The min- ⊕ ⊕ imal dimension ℓ is simply 1 ℓ = dim g+1 + 1 (2.2) 2 If we denote the subgroup generated by the grade zero subalgebra g0 as H, then the quotient G (2.3) H SL(2, R) × is a para-quaternionic symmetric space in the terminology of [19]. Our goal in this paper is to complete the construction of the minimal unitary representations of all such non-compact groups by quantization of their quasiconformal realizations. Remarkably, the para-quaternionic symmetric spaces arose earlier in the classification [21] of infinite dimensional nonlinear quasi- superconformal Lie algebras that contain the Virasoro algebra as a subalgebra 4. Below we list all simple noncompact groups G of this type and their subgroups H [21]: 4These infinite dimensional non-linear algebras were proposed as symmetry algebras that unify perturbative (Virasoro) and non-perturbative (U-duality) symmetries [22].

4 G H SU(m,n) U(m 1,n 1) − − SL(n, R) GL(n 2, R) − SO(n,m) SO(n 2,m 2) SU(1, 1) − − × SO∗(2n) SO∗(2n 4) SU(2) − × Sp(2n, R) Sp(2n 2, R) − E6(6) SL(6, R) E6(2) SU(3, 3) E6(−14) SU(5, 1) E7(7) SO(6, 6) ∗ E7(−5) SO (12) E7(−25) SO(10, 2) E8(8) E7(7) E8(−24) E7(−25) F4(4) Sp(6, R) G2(2) SU(1, 1)

The minimal unitary representations of the exceptional groups (F4, E6, E7, E8) and of SO(n, 4) as well as the corresponding quasiconformal realizations were given in [16, 15, 17].

3 Uniﬁed Construction of the minimal unitary realizations of non-compact groups with H simple or Abelian.

Consider the 5-graded decomposition of the Lie algebra g of G g−2 g−1 g0 ∆ g+1 g+2 ⊕ ⊕ ⊕ ⊕ ⊕ Let J a denote generators of the Lie algebra g0 of H

a b ab c J , J = f cJ (3.1a) h i where a, b, ... = 1, ...D and let ρ denote the symplectic representation by which g0 acts on g±1 a α a α β a α a α β [J , E ] = (λ ) βE [J , F ] = (λ ) βF (3.1b) where Eα, α, β, .. = 1, .., N = dim(ρ) are generators that span the subspace g−1 Eα , Eβ = 2ΩαβE (3.1c) h i 5 and F α are generators that span g+1

F α , F β = 2ΩαβF (3.1d) h i and Ωαβ is the symplectic invariant “metric” of the representation ρ. The negative grade generators form a Heisenberg subalgebra since

[Eα, E] = 0 (3.1e) with the grade -2 generator E acting as its central charge. Similarly the positive grade generators form a Heisenberg algebra with the grade +2 generator F acting as its central charge. The remaining nonvanishing commutation relations of g are

α α F α = [Eα , F ] [∆, E ]= E α −α α α [∆, F ]= F E = [E , F ] (3.1f) α β αβ αβ a [∆, E]= 2E E , F = Ω ∆+ ǫλa J − h i − [∆, F ] = 2F where ∆ is the generator that determines the ﬁve grading and ǫ is a parameter to be determined. We shall realize the generators using bosonic oscillators ξα satisfying the canonical commutation relations

ξα ,ξβ =Ωαβ (3.2) h i The grade -1, -2 generators and those of H can be realized easily as 1 1 E = y2 Eα = yξα J a = λa ξαξβ (3.3) 2 −2 αβ 1 2 where y, at this point, is an extra “coordinate” such that 2 y acts as the central charge of the Heisenberg algebra formed by the negative grade generators. Now there may exist different real forms of G with different subgroups H. For reasons that will become obvious we shall assume that a real form of G exists for which H is simple. We shall follow the conventions of [21] throughout this paper except for the occasional use of Cartan labeling of simple Lie algebras whenever we are not considering specific real forms. The quadratic Casimir operator of the Lie algebra g0 of H is

g0 = η J aJ b (3.4) C2 ab 6 where ηab is the Killing metric of H. The minimal realizations given in [16, 15, 17] and the results of [21] suggest an Ansatz for the grade +2 generator F of the form 1 F = p2 + κy−2 ( + C) (3.5) 2 C2 where p is the momentum conjugate to the coordinate y

[y,p]= i (3.6) and κ and C are some constants to be determined later. This implies then

F α = [Eα, F ]= ip ξα + κy−1 [ξα , ] C2 α −1 a α β α = ip ξ κy 2 (λ ) βξ Ja + Cρ ξ (3.7) − h i where Cρ is the eigenvalue of the second order Casimir of H in the representation ρ.5 We choose the normalization of the representation matrices λ as in [20, 21]

a,αβ γ a,γα β Cρ αβ γ βγ α γα β λ λa δ λ λa δ = Ω δ δ 2Ω δ δ +Ω δ δ , (3.8) − −N + 1 − which implies

λa λβγ = C δγ (3.9) αβ a − ρ α The unknown constants in our Ansatz will be determined by requiring that generators satisfy the commutation relations (3.1) of the Lie algebra g. We ﬁrst consider commutators of elements of g1 and g−1

α β αβ β α α β E , F = i (yp)Ω ξ ξ + κ ξ , ξ , 2 (3.10) h i − h h C ii which, upon using the identity,

[ξα , ]= 2 (λa)α ξβJ C ξα (3.11) C2 − β a − ρ leads to

α β αβ 3κCρ 1 α β β α a αβ E , F = ∆Ω + ξ ξ + ξ ξ 6κ (λ ) Ja (3.12) h i − 1+ N − 2 − 5Note that the indices α, β, .. are raised and lowered with the antisymmetric symplectic αβ − βα αβ α α αβ β metric Ω = Ω that satisﬁes Ω Ωγβ = δβ and V = Ω Vβ , and Vα = V Ωβα. In α α particular, we have V Wα = −VαW .

7 where ∆ = i (yp + py). Now the bilinears ξαξβ + ξβξα generate the Lie − 2 algebra of cN/2 (sp (N)) under commutation. Hence for those Lie algebras 0 g whose subalgebras g are diﬀerent from cN/2 closure requires that the coeﬃcient of the second term vanish 3κC 1 ρ = 0 (3.13) 1+ N − 2 0 For Lie algebras g whose subalgebras g are of type cN/2 we have

(λ )αβ J a ξαξβ + ξβξα a ≈ Hence we do not get any constraints on κ from the above commutation relation. Next, let us compute the commutator

α β κ α β β α α β 2 αβ F , F = 2 ξ ξ , 2 + ξ [ξ , 2]+ κ [ξ , 2] , ξ , 2 p Ω h i y − h C i C h C h C ii− (3.14) Using (3.8), (3.11) we write the terms linear in κ on the right hand side as

κ α β β α 2 ξ ξ , 2 + ξ [ξ , 2] = y − h C i C κ αβ α a β β a α γ 2 CρΩ + 2 ξ (λ ) γ ξ (λ ) γ ξ Ja . (3.15) y − The terms quadratic in κ on the right hand side gives

α β 12κCρ α a β β a α γ 2 αβ κ [ξ , 2] , ξ , 2 = ξ (λ ) γ ξ (λ ) γ ξ Ja κCρ Ω h C h C ii N + 1 − − αβ βα β b a b a c a α b µ ν +4κ 3 λ λ JaJb 2 λ λ JaJb + fab (λ ) µ λ ξ ξ Jc − ν Using these two expressions above (3.14) becomes

2 α β 1 2 1 κ 2 κ αβ F , F = 2 p + 2 Cρ Cρ Ω h i − 2 y 2 − 2 4κ α a β β a α γ + 2 ξ (λ ) γ ξ (λ ) γ ξ Ja y − 2 αβ βα β 4κ b a b a c a α b µ ν + 2 3 λ λ JaJb 2 λ λ JaJb + fab (λ ) µ λ ξ ξ Jc y − ν (3.16)

8 Now the right hand side of (3.16) must equal 2ΩαβF with 1 F = p2 + κy−2 ( + C) 2 C2 per our Ansatz. Contracting the right hand side of (3.16) with Ωβα we get

1 1 N p2 + κ2C2 κC κ 16 + 20κi ℓ2 4κC (3.17) − y2 ρ − ρ − y2 − ρ − adj C2 where iρ is the Dynkin index of the representation ρ of H and Cadj is the eigenvalue of the second order Casimir in the adjoint of H. To obtain this result one uses the fact that

λa λb,αβ = i ℓ2ηab αβ − ρ where ℓ is the length of the longest root of H.6 Using

C = ℓ2h∨ (3.18) adj − where h∨ is the dual Coxeter number of g0 subalgebra of g, the closure then requires 8 + 10κi ℓ2 + 2κh∨ℓ2 = N (3.19) − ρ Equations (3.13) and (3.19), combined with

N i ℓ2 = C (3.20) ρ D ρ imply h∨ 3D = (N + 8) 5 (3.21) iρ N(N + 1) − The validity of the above expression can be veriﬁed explicitly by com- paring with Table 1 of [21], relevant part of which is collected in Table 1 for convenience. Furthermore, it was shown in [21] that all the groups and the corresponding symplectic representations listed in the above table satisfy the equation D 3D h∨ = 2i + 1 (3.22) ρ N N(1 + N) − 6The length squared ℓ2 of the longest root is normalized such that it is 2 for the simply NCρ 4 2 laced algebras, 4 for Bn,Cn and F and 6 for G . The iρ,Cρ and ℓ are related by iρ = Dℓ2 where D = dim(H) = dim(g0).

9 0 ∨ g D h N = dimρ iρ 1 cn n(2n + 1) n + 1 2n 2 a5 35 6 20 3 d6 66 10 32 4 e7 133 18 56 6 5 c3 21 4 14 2 a1 3 2 4 5

Table 1: The list of grade zero subalgebras g0 with dual Coxeter number h∨ that are simple and with irreducible action ρ on grade +1 subspace. iρ is the Dynkin index of the representation ρ. which was obtained as a consistency condition for the existence of certain class of inﬁnite dimensional nonlinear quasi-superconformal algebras. Com- paring this equation with the equation (3.21) we see that they are consistent with each other if 3N (N + 1) D = (3.23) N + 16 Requirement of [F , F α] = 0 leads to the condition ξα ( + C) + ( + C) ξα + κ [ , [ξα , ]] = 0 (3.24) C2 C2 C2 C2 Using (3.11) and [ , J a] = 0 we arrive at C2 α α a α β 2 ξ ( 2 + C)+2(1 κCρ) Cρξ +2(1 κCρ) (λ ) β ξ Ja C − − α a b β 4 κ λ λ ξ JbJa = 0 − β (3.25) In order to extract restrictions on g implied by the above equation we con- γ tract it with ξ Ωγα and obtain h∨ D = (N 8) + 1 . (3.26) iρ N(N + 1) − It agrees with (3.21) provided (3.23) holds true. Making use of N = 2(g∨ 2) − where g∨ is the dual Coxeter number of the Lie algebra g and (3.23) we obtain (g∨ + 1) (5g∨ 6) dim (g)=2+2N +1+dim g0 =1+2(N +1)+ D = 2 − g∨ + 6 (3.27)

10 g dim(g) g∨ Eqtn. (3.27) holds ? 2 an n + 2n n + 1 for a1 and a2 only 2 bn 2n + n 2n 1 no 2 − cn 2n + n n + 1 no d 2n2 n 2n 2 for d only n − − 4 e6 78 12 yes e7 133 18 yes e8 248 30 yes f4 52 9 yes g2 14 4 yes Table 2: Dimensions and dual Coxeter numbers of simple Lie algebras. In order for the Lie algebra to admit a non-trivial 5-graded decomposition its dimension must be greater than 6. This rules out sl(2) for which (3.27) also holds.

Equation (3.24) and the requirement of the right hand side of (3.16) to equal to 2ΩαβF imply restriction on matrices λa for which (3.21) and (3.26) are only necessary conditions. We expect these conditions to be derivable from the identities satisﬁed by the corresponding Freudenthal triple systems [36] that underlie the quasiconformal actions and the minimal realizations [14, 17]. By going through the list of simple Lie algebras [25] collected for convenience in Table 2 we see that the equation (3.23) is valid only for the Lie algebras of simple groups A1, A2, G2, D4, F4,E6, E7 and E8. For A1 our realization reduces simply to the conformal realization. With the exception of D4, what these groups have in common is the fact that their subgroups H are either simple or one dimensional Abelian as expected by the consistency with our Ansatz. The reason our Ansatz also covers the case of D4 has to 0 do with its unique properties. The subalgebra g of d4 is the direct sum of three copies of a1 g0 = a a a 1 ⊕ 1 ⊕ 1 The eigenvalues of the quadratic Casimirs of these subalgebras a1 as well as their Dynkin indices in the representation ρ coincide as required by the consistency with our Ansatz. These groups appear in the last row of the so- called Magic Triangle [26] which extends the Magic Square of Freudenthal, Rozenfeld and Tits [28]. There is, in addition, an inﬁnite family of non-compact groups for which H is simple, namely the noncompact symplectic groups Sp(2n + 2, R) with

11 dim ρ = 2n and H = Sp(2n, R). However, as remarked above, the constraint (3.13) and hence the equation (3.23) do not follow from our Ansatz for the symplectic groups. The quartic invariant becomes degenerate for symplectic groups and the minimal unitary realizations reduce to free boson construction of the singleton representations for these groups as will be discussed in the next section. The minimal unitary realizations of noncompact groups appearing in the Magic Triangle [28, 26] can be obtained by consistent truncation of the minimal unitary realizations of the groups appearing in its last row [15, 14, 17]. We should stress that there are different real forms of the groups appearing in the Magic Square or its straightforward extension to the Magic Triangle. Different real forms in our unified construction correspond to different hermiticity conditions on the bosonic oscillators ξα [15]. After specifying the hermiticity properties of the oscillators ξα one goes to a Hermitian (anti- hermitian) basis of the Lie algebra g with purely imaginary (real) structure constants to calculate the Killing metric which determines the real form corresponding to the minimal unitary realization. The quadratic Casimir operator of the Lie algebra constructed in a unified manner above is given by

a 2 Cρ 1 2 Cρ α β β α 2 (g)= J Ja + ∆ + EF + F E Ωαβ E F + F E C N + 1 2 − N + 1 (3.28) which, upon using (3.13) and the following identities

1 2 a 3 ∆ + EF + F E = κ (J Ja + C) 2 − 8 (3.29) α β β α a N Ωαβ E F + F E = 8 κJ Ja + + κCρN 2 that follow from our Ansatz, reduces to a c-number 8κC 3 C N C κC2N (g)= C ρ 1 ρ ρ ρ 2 N + 1 4 N + 1 2 N + 1 N + 1 C − − − − (3.30) C (N + 4) (5 N + 8) = ρ (using eq.(3.13)) − 36 N + 1 as required by irreducibility. We should note that this result agrees with explicit calculations for the Lie algebras of the Magic Square in [15]. In the normalization chosen there κ = 2 and hence 12Cρ = N + 1. Then, using N = 2g∨ 4 we get − 1 (g)= 5g∨ 6 g∨. (3.31) C2 −108 −

12 4 The minimal unitary representations of Sp (2n +2, R)

The Lie algebra of Sp (2n + 2, R) has a 5-grading of the form sp (2n + 2, R)= E Eα (sp (2n, R) ∆) F α F (4.1) ⊕ ⊕ ⊕ ⊕ ⊕ α α 1 2 where E = yξ , and E = 2 y . Generators of the grade zero subalgebra g0 = sp (2n, R) are given simply by the symmetrized bilinears ( modulo normalization) 2 (λ )αβ J a = ξαξβ + ξβξα (4.2) − a which is simply the singleton ( metaplectic) realization of sp(2n, R). The quadratic Casimir of sp (2n, R) in the singleton realizaton is simply a c- number. As stated in the previous section the constraint equation (3.13) that follow from the commutation relations [Eα, F β] can not be imposed in the case of symplectic Lie algebras Sp (2n + 2, R). However the equation (3.24) that follows from our Ansatz requires that 2 + C = 0 or κ = 0 for R C 1 2 the symplectic Lie algebras Sp (2n + 2, ). In other words F = 2 p . Thus F α = [Eα, F ]= ipξα (4.3) Eα, F β = i (yp)Ωαβ ξβξα h i − i 1 = ∆ (ξαξβ + ξβξα) (4.4) −2 − 2 Thus the minimal unitary realization of the symplectic group Sp(2n + 2, R) obtained by quantization of its quasiconformal realization is simply the singleton realization in terms bilinears of the 2n + 2 oscillators (annihilation and creation operators) (ξα,y,p). The quadratic Casimir of sp (2n + 2, R) is also a c-number. That the quadratic Casimir is a c-number is only a necessary requirement for the irreducibility of the corresponding represenation. For the above singleton realization the entire Fock space of all the oscillators decompose into the direct sum of the two inequivalent singleton representations that have the same eigenvalue of the quadratic Casimir. They are both unitary lowest weight representations. By choosing a deﬁnite polarization one can deﬁne n + 1 annihilation operators 1 a0 = (y + ip) √2

1 i n+i ai = (ξ + iξ ) i = 1, 2, .., n √2

13 and n + 1 creation operators 1 a0 = (y ip) √2 − 1 ai = (ξi iξn+i) √2 − in terms of the (n+1) coordinates and (n+1) momenta. The vacuum vector 0 annihilated by all the annihilation operators | i a 0 = a 0 = 0 0| i i| i is the lowest weight vector of the “scalar” singleton irrep of Sp(2n + 2, R) and (n+1) vectors a0 0 , ai 0 | i | i form the lowest K-vector of the other singleton irrep of Sp(2n + 2, R). In other words the lowest K vector of the scalar singleton is an SU(n + 1) scalar, while the lowest K-vector of the other singleton irrep is a vector of SU(n+1) subgroup of Sp(2n+2, R). Both lowest K-vectors carry a nonzero U(1) charge. The reason for the reduction of the minimal unitary realizations of the Lie algebras of symplectic groups Sp(2n + 2, R) to bilinears, and hence to a free boson construction, is the fact that there do not exist any nontrivial quartic invariant of Sp(2n, R) deﬁned by an irreducible symmetric tensor in the fundamental representation 2n. We have only the skew symmetric symplectic invariant tensor Ωαβ in the fundamental representation, which when contracted with ξαξβ gives a c-number. In the light of the above results one may wonder how the quasi-conformal realization of sp (2n + 2, R) can be made manifest. Before quantisation we have 2n + 1 coordinates X = (Xα,x) on which we realize sp (2n + 2, R):

(X)= I (Xα) x2 (4.5) N 4 − α since I4 (X ) = 0. With the “twisted” difference vector defined as [14] δ (X, Y) = (Xα Y α,x y + X,Y ) (4.6) − − h i The equation defining the generalized lightcone

(δ (X, Y)) = 0 N then reduces to x y + X,Y = 0 (4.7) − h i

14 where X,Y = Ω XαY β. By reinterpreting the coordinates (Xα,x) and h i αβ (Y α,y) as projective coordinates in 2n+2 dimensional space

ξ0 x = ξn+1 ξα Xα = ξn+1 η0 y = ηn+1 ηα Y α = ηn+1 the above equation for the light cone can be written in the form

ξ0ηn+1 η0ξn+1 +Ω ξαηβ = 0 − αβ which is manifestly invariant under Sp(2n + 2, R).

5 Minimal unitary realizations of the quasiconformal groups SO(p +2,q + 2)

In our earlier work [17] we constructed the minimal unitary representations of SO (d + 2, 4) obtained by quantization of their realizations as quasiconformal groups. That construction carries over in a straightforward manner to the other real forms SO (p + 2,q + 2) which we give in this section. They were studied also in [34] using the quasiconformal approach and in [35] by other methods. Now the relevant subgroup for the minimal unitary realization is

SO(p,q) SO(2, 2) SO(p + 2,q + 2) (5.1) × ⊂ where SO(2, 2) = Sl(2, R) Sl(2, R)= Sp(2, R) Sp(2, R) (5.2) × × and one of factors above can be identiﬁed with the distinguished Sl (2, R) subgroup. The relevant 5-grading of the Lie algebra of SO(p + 2,q + 2) is then given as

so(p + 2,q +2) = g−2 g−1 (so(p,q) sp (2, R) ∆) g+1 g+2 (5.3) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

15 where grade 1 subspaces transform in the [(p + q), 2] dimensional repre- ± sentation of SO(p,q) Sl(2, R). µ × (p,q) Let X and Pµ be canonical coordinates and momenta in R :

µ µ [X , Pν ]= iδν (5.4) Also let x be an additional “cocycle” coordinate and p be its conjugate momentum: [x,p]= i (5.5) They are taken to satisfy the following Hermiticity conditions:

µ † ν † µν † † (X ) = ηµνX (Pµ) = η Pν p = p x = x (5.6) where ηµν is the SO(p,q) invariant metric. The subgroup H of SO(p+2,q + 2) is SO(p,q) Sp(2, R) whose generators we will denote as (M , J±, J ). × J µν 0 The grade 1 generators will be denoted as (U , V µ) and the grade 2 − µ − generator as K−. The generators of H, its 4-th order invariant and the I4 negative grade generators are realized as follows:

ρ ρ Mµν = iηµρX Pν iηνρX Pµ 1 µ µ − J0 = (X Pµ + PµX ) µ µ 2 Uµ = xPµ V = xX J = XµXν η 1 − µν K = x2 µν (5.7) − 2 J+ = PµPνη = (XµXνη ) (P P ηµν ) + (P P ηµν ) (XµXνη ) I4 µν µ ν µ ν µν (XµP ) (P Xν ) (P Xµ) (XνP ) − µ ν − µ ν where ηµν is the ﬂat metric with signature (p,q). It is easy to verify that the generators Mµν and J0,± satisfy the commutation relations of so (p,q) sp (2, R) ⊕ [Mµν , Mρτ ]= ηνρMµτ ηµρMντ + ηµτ Mνρ ηντ Mµρ − − (5.8) [J , J±]= 2iJ± [J−, J ] = 4iJ 0 ± + 0 µ under which coordinates X and momenta Pµ transform as SO (p,q) vectors and form doublets of the symplectic group Sp(2, R)J :

µ µ µ µ µν [J0,V ]= iV [J−,V ] = 0 [J+,V ]= 2iη Uν − ν − (5.9) [J0,Uµ]=+iUµ [J−,Uµ] = 2iηµν V [J+,Uµ] = 0

The generators in the subspace g−1 g−2 form a Heisenberg algebra ⊕ µ µ [V ,Uν ] = 2iδ ν K− . (5.10)

16 with K− playing the role of central charge. Using the quartic invariant we deﬁne the grade +2 generator as 1 1 (p + q 2)2 + 3 K = p2 + + − (5.11) + 2 4 y2 I4 2 Then the grade +1 generators are obtained by commutation relations

V˜ µ = i [V µ, K ] U˜ = i [U , K ] (5.12) − + µ − µ + which explicitly read as follows

µ µ 1 −1 λ ρ λ ρ µν V˜ = pX + x Pν X X + X X Pν η ηλρ 2 1 −1 µ ν ν ν ν µ x (X (X Pν + Pν X ) + (X Pν + PνX ) X ) − 4 (5.13) 1 U˜ = pP x−1 (Xν P P + P P Xν) η ηλρ µ µ − 2 λ ρ λ ρ µν 1 + x−1 (P (Xν P + P Xν) + (Xν P + P Xν ) P ) . 4 µ ν ν ν ν µ Then one ﬁnds that the generators in g+1 g+2 subspace form an Heisenberg ⊕ algebra as well

µ µ µ µ V˜ , U˜ν = 2iδ ν K+ V = i V˜ , K− Uµ = i U˜µ, K− . (5.14) h i h i h i Commutators g−1, g+1 close into g0 as follows µ ν µν Uµ, U˜ν = iηµν J− V , V˜ = iη J+ h i h i µ µρ µ V , U˜ν = 2η Mρν + iδ ν (J0 + ∆) (5.15) h i ν νρ ν Uµ, V˜ = 2η Mµρ + iδ µ (J0 ∆) h i − − where ∆ is the generator that determines the 5-grading 1 ∆= (xp + px) (5.16) 2 such that [K−, K ]= i∆ [∆, K±]= 2iK± (5.17) + ±

[∆,U ]= iU [∆,V µ]= iV µ ∆, U˜ = iU˜ ∆, V˜ µ = iV˜ µ µ − µ − µ µ h i h i (5.18)

17 R The quadratic Casimir operators of subalgebras so (p,q), sp (2, )J of grade R zero subspace and sp (2, )K generated by K± and ∆ are

M M µν = 2 (p + q) µν −I4 − 2 1 2 J−J + J J− 2 (J ) = + (p + q) + + − 0 I4 2 (5.19) 1 2 1 1 2 K−K + K K− ∆ = + (p + q) + + − 2 4I4 8 Note that they all reduce to modulo some additive and multiplicative I4 constants. Noting also that

µ µ µ µ UµV˜ + V˜ Uµ V U˜µ U˜µV = 2 4 + (p + q) (p + q + 4) (5.20) − − I we conclude that there exists a family of degree 2 polynomials in the en- veloping algebra of so (p + 2,q + 2) that degenerate to a c-number for the minimal unitary realization, in accordance with Joseph’s theorem [18]:

µν 2 1 2 Mµν M + κ1 J−J+ + J+J− 2 (J0) + 4κ2 K−K+ + K+K− ∆ − − 2 1 µ µ µ µ (κ1 + κ2 1) UµV˜ + V˜ Uµ V U˜µ U˜µV − 2 − − − 1 = (p + q) (p + q 4 (κ + κ )) 2 − 1 2 (5.21)

The quadratic Casimir of so (p + 2,q + 2) corresponds to the choice 2κ1 = 2κ2 = 1 in (5.21). Hence the eigenvalue of the quadratic Casimir for the − 1 minimal unitary representation is equal to 2 (p + q) (p + q + 4).

6 Minimal unitary realizations of the quasiconformal groups SO∗(2n + 4)

The noncompact group SO∗(2n +4) is a subgroup of SL(2n + 4, C) whose maximal compact subgroup is U(n + 2). We have the inclusions

SO∗(2n + 4) SU ∗(2n + 4) SL(2n + 4, C) (6.1) ⊂ ⊂ As a matrix group SU ∗(2n + 4) is generated by matrices U belonging to SL(2n + 4, C) that satisfy UJ = JU ∗ (6.2)

18 where J is a (2n + 4) (2n + 4) matrix that is antisymmetric × JT = J (6.3) − and whose square is the identity matrix

J2 = I (6.4) − The matrices U belonging to the subgroup SO∗(2n + 4) of SU ∗(2n + 4) satisfy, in addition, the condition

UU T = I (6.5)

∗ Thus SO (2n + 4) leaves invariant both the Euclidean metric δIJ and the complex structure J = J where I, J, .. = 1, 2, ...2n+4. Hence SO∗(2n+ IJ − JI 4) is also a subgroup of the complex rotation group SO(2n + 4, C). To obtain the 5-grading of the Lie algebra of SO∗(2n + 4) so as to construct its minimal unitary representation we need to consider its decomposition with respect to its subgroup

SO∗(2n) SO∗(4) SO∗(2n + 4) (6.6) × ⊂ where SO∗(4) = SU(2) SL(2, R) (6.7) × The distinguished SL(2, R) subgroup can then be identiﬁed with the factor SL(2, R) above. The corresponding 5-grading of the Lie algebra of SO∗(2n+ 4) is then

so∗(2n +4) = g−2 g−1 (so∗(2n) su(2) ∆) g+1 g+2 (6.8) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ where grade 1 subspaces transform in the [2n, 2] dimensional representa- ± tion of SO∗(2n) SU(2). Let Xµ and P be canonical coordinates and × µ momenta in R(2n): µ µ [X , Pν ]= iδν (6.9) They satisfy the Hermiticity conditions

µ † ν (X ) = Jµν X (6.10)

† µν (Pµ) = J Pν (6.11)

19 Let x be an additional “cocycle” coordinate and p be its conjugate momentum: [x,p]= i (6.12) as in the previous section. The subgroup H is now SO∗(2n) SU(2) whose × J generators are (Mµν , J±, J0). The grade 1 generators will be denoted as µ − (U , V ) and the grade 2 generator as K− as in the previous section. The µ − generators of H, its 4-th order invariant and the negative grade generators I4 are realized as follows: ρ ρ Mµν = iδµρX Pν iδνρX Pµ 1 µ µ − J0 = (X Pµ + PµX ) µ µ 2 Uµ = xPµ V = xX J = XµXµ 1 − K = x2 (6.13) − 2 J+ = PµPµ = (XµXµ) (P P ) + (P P ) (Xν Xν) I4 ν ν µ µ (XµP ) (P Xν ) (P Xµ) (XνP ) − µ ν − µ ν where δµν is the ﬂat Euclidean metric in 2n dimensions. It is easy to verify that the generators Mµν and J0,± satisfy the commutation relations of so∗ (2n) su (2) : ⊕ J

[Mµν , Mρτ ]= δνρMµτ δµρMντ + ηµτ Mνρ ηντ Mµρ − − (6.14) [J , J±]= 2iJ± [J−, J ] = 4iJ 0 ± + 0 under which coordinates Xµ (V µ) and momenta P µ (U µ) transform as vectors of SO∗(2n) and form doublets of SU(2):

µ µ µ µ ν [J0,V ]= iV [J−,V ] = 0 [J+,V ]= 2iU − − (6.15) [J0,Uµ]=+iUµ [J−,Uµ] = 2iVµ [J+,Uµ] = 0

The generators in the subspace g−1 g−2 form a Heisenberg algebra ⊕ µ µ [V ,Uν ] = 2iδ ν K− . (6.16) with K− playing the role of “~”. Using the quartic invariant we deﬁne the grade +2 generator as

1 1 4(n 1)2 + 3 K = p2 + + − (6.17) + 2 4 y2 I4 2 Then the grade +1 generators are obtained by commutation relations

V˜ µ = i [V µ, K ] U˜ = i [U , K ] (6.18) − + µ − µ +

20 which explicitly read as follows

µ µ 1 −1 λ ρ λ ρ µν V˜ = pX + x Pν X X + X X Pν η ηλρ 2 1 −1 µ ν ν ν ν µ x (X (X Pν + Pν X ) + (X Pν + PνX ) X ) − 4 (6.19) 1 U˜ = pP x−1 (Xν P P + P P Xν) η ηλρ µ µ − 2 λ ρ λ ρ µν 1 + x−1 (P (Xν P + P Xν) + (Xν P + P Xν ) P ) . 4 µ ν ν ν ν µ Then one ﬁnds that the generators in g+1 g+2 subspace form an Heisenberg ⊕ algebra as well

µ µ µ µ V˜ , U˜ν = 2iδ ν K+ V = i V˜ , K− Uµ = i U˜µ, K− . (6.20) h i h i h i Commutators g−1, g+1 close into g0 as follows µ ν µν Uµ, U˜ν = iηµν J− V , V˜ = iη J+ h i h i µ µρ µ V , U˜ν = 2η Mρν + iδ ν (J0 + ∆) (6.21) h i ν νρ ν Uµ, V˜ = 2η Mµρ + iδ µ (J0 ∆) h i − − where ∆ is the generator that determines the 5-grading 1 ∆= (xp + px) (6.22) 2 such that [K−, K ]= i∆ [∆, K±]= 2iK± (6.23) + ±

[∆,U ]= iU [∆,V µ]= iV µ ∆, U˜ = iU˜ ∆, V˜ µ = iV˜ µ µ − µ − µ µ h i h i (6.24) ∗ The quadratic Casimir operators of subalgebras so (2n), su (2)J of grade R zero subspace and sp (2, )K generated by K± and ∆ are M M µν = 2 (p + q) µν −I4 − 2 1 2 J−J + J J− 2 (J ) = + (p + q) + + − 0 I4 2 (6.25) 1 2 1 1 2 K−K + K K− ∆ = + (p + q) + + − 2 4I4 8

21 Note that they all reduce to modulo some additive and multiplicative I4 constants. Noting also that

µ µ µ µ UµV˜ + V˜ Uµ V U˜µ U˜µV = 2 4 + (p + q) (p + q + 4) (6.26) − − I we conclude that there exists a family of degree 2 polynomials in the envelop- ing algebra of so∗ (2n + 4) that degenerate to a c-number for the minimal unitary realization, in accordance with Joseph’s theorem [18]:

µν 2 1 2 Mµν M + κ1 J−J+ + J+J− 2 (J0) + 4κ2 K−K+ + K+K− ∆ − − 2 1 µ µ µ µ (κ1 + κ2 1) UµV˜ + V˜ Uµ V U˜µ U˜µV − 2 − − − = n (2n 4 (κ + κ 1)) − 1 2 − (6.27)

∗ The quadratic Casimir of so (2n + 4) corresponds to the choice 2κ1 = 2κ = 1 in (6.27). Hence the eigenvalue of the quadratic Casimir for the 2 − minimal unitary representation is equal to 2n (n + 2).

7 Minimal unitary realizations of the quasiconformal groups SU (n +1, m + 1)

The Lie algebra su (n + 1,m + 1) admits the following ﬁve graded decomposition with respect to its subalgebra su (n,m):

su (n + 1,m + 1) = 1 2(n + m) (su (n,m) u (1)) 2(n + m) 1 ⊕ ⊕ ⊕ ⊕ ⊕ It is realized by means of m + n pairs of creation and annihilation operators subject to the following Hermiticity condition:

p † pq p p (a ) = η aq [aq, a ]= δ q . (7.1)

Following the steps laid down in previous sections we deﬁne generators of H as bilinears in creation and annihilation operators 1 J p = apa δp ara (7.2a) q q − m + n q r Negative grade generators are 1 E = x2 Ep = xap E = xa (7.2b) 2 q q

22 The quartic invariant is related to the quadratic Casimir of H simply I4 2 (m + n) 1 = J p J q + (m + n)2 1 (7.2c) I4 m + n 1 q p 2 − − where the additive constant was determined such that 1 1 1 F = p2 + (7.2d) 2 4 x2 I4 The positive grade g+1 generators are then found by commuting F with generators of g−1

F p = i [Ep, F ] F = i [E , F ] (7.2e) − q − q The u(1) generator of grade 0 subalgebra is also bilinear in oscillators 1 U = (apa + a ap) (7.2f) 2 p p Quadratic Casimir of the algebra in this realization reduces to a c-number 1 1 1 1 m + n + 2 = J p J q + ∆2 (EF + F E) U 2 C2 − 6 q p 12 − 6 − 12 m + n i (E F p + F pE F Ep EpF ) (7.3) − 12 p p − p − p 1 = (m + n + 2) (m + n + 1) 24 Positive and negative grades generators transform in the (n+m)+1 (n + m)−1 ⊕ ⊕ 10 representation of H and satisfy

[J p , J s ]= δs J p δp J s q t q t − t q 1 [J p , Es]= δs Ep δp Es q q − n + m q 1 [J p , F s]= δs F p δp F s q q − n + m q (7.4) 1 [J p , E ]= δp E + δp E q s − s q n + m q s 1 [J p , F ]= δp F + δp F q s − s q n + m q s

[U, Ep]= Ep [U, F p]= F p [U, E ]= E [U, F ]= F (7.5) p − p p − p

23 [∆, Ep]= iEp [∆, E ]= iE [∆, F p]= iF p [∆, F ]= iF (7.6) − p − p p p

[∆, F ] = 2iF [∆, E]= 2iE [E, F ]= i∆ (7.7) − The remaining non-zero commutation relations are as follows:

q q [Ep, E ] = 2E [Fp, F ] = 2F (7.8)

m + n + 2 [Ep, F ] = 2iJ p + iδp U δp ∆ q q m + n q q − (7.9) m + n + 2 [E , F q]= 2iJ q iδq U δp ∆ p − p − m + n p − q

[F, E ]= iF [F, Ep]= iF p [E, F ]= iE [E, F p]= iEp (7.10) p − p − p p 8 Minimal unitary realizations of the quasiconformal groups SL (n +2, R)

The construction of the minimal unitary realization of the quasiconformal algebra sl(n + 2, R) traces the same steps as in the previous section. The ﬁve-graded decomposition is as follows

sl (n + 2, R) = 1 (n n˜) (gl (n, R) ∆) (n n˜) 1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Since n n˜ is a direct sum of two inequivalent self-conjugate vector repre- ⊕ µ sentations of gl (n, R), we use coordinates X and momenta Pµ as oscillator generators, where µ = 1,...,n, with canonical commutation relations:

µ µ [X , Pν ]= iδ ν (8.1)

Generators of gl (n, R) i µ = (XµP + P Xµ) (8.2) L ν 2 ν ν have the following commutation relations

[ µ , τ ]= δτ µ δµ τ (8.3) L ν L ρ νL ρ − ρL ν

24 The one-dimensional center of the reductive algebra gl (n, R) is spanned by

n U = µ (8.4) L µ µX=1

The quadratic Casimir of gl (n, R) is given simply as a trace of 2: L n (gl (n, R)) = µ ν = (XµP ) (P Xν) (8.5) C2 L νL µ 4 − µ ν Generators of the negative grades 1 E = x2 Eµ = xXµ E = xP (8.6) 2 ν ν form the Heisenberg algebra

µ µ µ [E , Eν ] = (2iE) δ ν [E, E ] = 0 ν µ (8.7) [E , E ]=0 [E, Eµ]=0 [Eν, Eµ] = 0 The generator of grade +2 subspace takes on the familiar form

1 1 1 1 1 1 n2 1 F = p2 + = p2 + − (XµP ) (P Xν ) (8.8a) 2 2 x2 I4 2 2 x2 4 − µ ν and leads to the following grade +1 generators 1 1 F µ = i [xXµ, F ]= pXµ (Xµ (P Xν) + (Xν P ) Xµ) (8.8b) − − 2 x ν ν

1 1 F = i [xP , F ]= pP + (P (P Xν ) + (Xν P ) P ) (8.8c) µ − µ µ 2 x µ ν ν µ They form the dual Heisenberg algebra

µ µ µ [F , Fν ] = (2iF ) δ ν [F, F ] = 0 ν µ (8.9) [F , F ]=0 [F, Fµ]=0 [Fν , Fµ] = 0

Subspaces g±1 transform under gl (n, R) as n n˜ each: ⊕ µ ρ ρ µ µ µ [ ν, E ]= δ νE [ ν, Eρ]= δ ρEν L L − (8.10) [ µ , F ρ]= δρ F µ [ µ , F ]= δµ F L ν ν L ν ρ − ρ ν Other cross grade commutation relations read

[E, F µ]= iEµ [E, F ]= iE [F, Eµ]= iF µ [F, E ]= iF (8.11) µ µ − µ − µ

25 [E, F ]= i∆ [∆, E]= 2iE [∆, F ]=+2iF (8.12) −

µ µ [∆, E ]= iE [∆, Eµ]= iEµ µ − µ − (8.13) [∆, F ]=+iF [∆, Fµ]=+iFµ

µ ν [E , F ]=0 [Eµ, Fν ] = 0 (8.14)

µ µ µ [E , Fν ] = 2 ν + δ ν (U + i∆) L (8.15) [E , F ν ] = 2 ν + δν (U i∆) µ L µ µ − A short calculation veriﬁes that the quadratic Casimir of sl (n + 2, R) is

µ ν 1 µ ν 1 2 C2 = ν µ + ( µ) ( ν ) ∆ + (EF + F E) L L 2 L L − 2 (8.16) 1 + (EµF + F Eµ F µE E F µ) 2 µ µ − µ − µ When evaluated on the quasi-conformal realization it reduces to c-number: 1 C = (n + 2) (n + 1) (8.17) 2 −4 Quadratic Casimir is just an element of the Joseph ideal as follows from relations below 1 3 1 µ ν + ( µ ) ( ν )= + (3 2n (n 1)) (8.18) L νL µ 2 L µ L ν 2I4 8 − −

1 1 3 ∆2 + (EF + F E)= (8.19) −2 2I4 − 8 1 1 (EµF + F Eµ F µE E F µ)= 2 n + (8.20) 2 µ µ − µ − µ − I4 − 2 Namely, any linear combination of the above three expressions collapses to a c-number provided coeﬃcients are matched as to cancel all . I4

26 9 Minimal Unitary Realizations of Lie Superalge- bras

In this section we will extend the construction of the minimal unitary representations of Lie groups obtained by quantization of their quasi conformal realizations to the construction of the minimal representations of Lie superalgebras. In analogy with the Lie algebras we consider 5-graded simple Lie superalgebras g−2 g−1 g0 ∆ g+1 g+2 (9.1) B ⊕ F ⊕ ⊕ B ⊕ F ⊕ B where g±2 are 1-dimensional subspaces each, and g−2 ∆ g+2 form sl(2 , R). ⊕ ⊕ In this paper we restrict ourselves to Lie superalgebras whose grade 1 ± generators are all odd ( exhaustively). Let J a denote generators of g0

a b ab c J , J = f cJ (9.2) h i and let ρ denote the irreducible orthogonal representation with a deﬁnite Dynkin index by which g0 acts on g±1

a α aα β a α aα β [J , E ]= λ βE [J , F ]= λ βF (9.3) where Eα are odd generators that span g−1

α β αβ E , E =Ωs E (9.4) n o and F α generators that span g+1

α β αβ F , F =Ωs F (9.5) n o αβ and Ωs is now a symmetric invariant tensor. Hence negative (positive) grade subspace form a super Heisenberg algebra. Due to 5-graded structure we can impose F α = [Eα , F ] Eα = [E , F α] (9.6) Now we realize the generators using anti-commuting covariant oscillators ξα

α β αβ ξ ,ξ =Ωs (9.7) n o plus an extra bosonic coordinate y and its conjugate momentum p. The non-positive grade generators take the form 7 1 1 E = y2 Eα = yξα J a = λa ξαξβ (9.8) 2 −2 αβ 7We are following conventions of [21] for superalgebras as well.

27 The quadratic Casimir of g0 is taken to be

g0 = η J aJ b (9.9) C2 ab and the grade +2 generator F is assumed to be of the form 1 F = p2 + κy−2 ( + C) (9.10) 2 C2 for some constants κ and C to be determined later. Commuting E with F α we obtain F α = ip ξα + κy−1 [ξα , ] (9.11) C2 By inspection we have

[ξα , ]= 2 (λa)α ξβJ + C ξα (9.12) C2 − β a ρ and we shall use the following Ansatz [21]

a β α a βα Cρ αβ β α α β (λ ) γ(λa) δ + (λ ) (λa)γδ = Ωs Ωsγδ + δ γ δ δ 2δ γδ δ N 1 − − (9.13) to calculate the remaining super commutation relations. For the anticommutators of grade +1 generators with grade -1 generators we get

α β αβ β α α β E , F = i (yp)Ωs + ξ ξ + κ ξ , ξ , 2 (9.14) n o n h C io 3κC 1 Eα, F β = Ωαβ∆ 6κ (λa)βα J + ρ ξβξα ξαξβ . − s − a N 1 − 2 − n o − Now the bilinears ξβξα ξαξβ on the right hand side generate the Lie − algebra so(N). Therefore, for those Lie superalgebras whose grade zero subalgebras g0 are diﬀerent from so(N) we must impose the constraint:

3κC 1 ρ (9.15) N 1 − 2 − For the anticommutators F α, F β we get α β 2 αβ F , F = p Ωs n o − κ α β β α α β 2 ξ ξ , 2 + ξ [ξ , 2] κ [ξ , 2] , ξ , 2 (9.16a) − y h C i C − n C h C io

28 2 α β αβ p k 1 2 1 αβ F , F = 2F Ωs = 2 + 2 κCρ + Cρ Ωs n o − − 2 x 2 2 κ α β β α γ a αβ b a 2 4 ξ (λa) γ + ξ (λa) γ ξ J + 12κ (λaλb) J J − x − βα b a α β δ γ ab c +8κ (λaλb) J J 4κ(λa) δ(λb) γξ ξ f cJ − (9.16b)

Taking the Ωs trace we obtain N = 8 10κi ℓ2 + 2κC 2C = κC2 + C (9.17) − ρ adj ρ ρ Taking into account that

2 iρℓ N 2 ∨ = Cadj =+ℓ h (9.18) Cρ D and using (9.15) we obtain the following constraint equation for super quasiconformal algebras, whose grade zero algebras are diﬀerent from so(N):

h∨ 3D =5+ (N 8) . (9.19) i N(N 1) − ρ − Now we also have κ [F, F α]= (( + C) ξα + ξα ( + C)+ κ [ , [ξα, ]]) (9.20) x3 C2 C2 C2 C2 Hence the constraint imposed by the commutation relation [F, F α]=0 is

2ξα ( + C) C ξα + (2 + 4κC ) (λ )α κC2ξα 4κ(λ λ )α ξβJ bJ a = 0 C2 − ρ ρ a β − ρ − a b β (9.21) γ which, upon contraction with ξ Ωsγα leads to the following condition h∨ D = 1+ (N + 8) . (9.22) i − N (N 1) ρ − These two conditions (9.19) and (9.22) agree provided 3N (N 1) D = − (9.23) 16 N − which is also the condition for them to agree with the equation D 3D h∨ = 2i + + 1 (9.24) ρ N N(1 N) − 29 which was obtained as a consistency condition for the existence of certain class of inﬁnite dimensional nonlinear superconformal algebras [21, 20]. Looking at the tables of simple Lie superalgebras [21, 20] consistent with our Ansatz we ﬁnd the following simple Lie algebras g0 and their irreps of dimension N that satisfy these conditions:

g0 DN b3 21 8s (9.25) g2 14 7 These solutions correspond to the Lie superalgebras f(4) with even subalgebra b sl(2, R) and g(3) with even subalgebras g sl(2, R). 3 ⊕ 2 ⊕ We should note that the real forms with an even subgroup of the form H SL(2, R), with H simple, admit unitary representations only if H is × compact.

10 Minimal representations of OSp (N 2, R) | and D(2, 1; α)

10.1 OSp(N 2, R) | For the Lie superalgebras osp(N 2, R) the constraint equation (9.15) does | not follow from the commutation relations and hence must not be imposed. In this case the bilinears ξβξα ξαξβ generate the Lie algebra of the even − subgroup SO(N) ( super analog of Sp(2n, R). This realization corresponds to the singleton representation of SO(N) and its quadratic Casimir is a c-number. As a consequence of this the Jacobi identities require either that we set κ = 0 as in the case of the minimal realization of Sp(2n + 2, R). Hence the minimal realization reduces to a realization in terms of bilinears of fermionic and bosonic oscillators. Thus the minimal unitary representations of OSp(N 2, R) are simply the supersingleton representations. The singleton | supermultiplets of OSp(N 2, R) were studied in [27]. | 10.2 D(2, 1; σ) There is a one parameter family of simple Lie superalgebras of the same dimension that has no analog in the theory of ordinary Lie algebras. It is the family D (2, 1; σ). The real forms of interest to us that admit unitary representations has the even subgroup SU(2) SU(2) SL(2, R). It has a × × ﬁve grading of the form D (2, 1; σ)= 1 (2, 2) (su (2) su (2) ∆) (2, 2) 1 (10.1) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

30 Let Xα,α˙ be 4 fermionic oscillators with canonical anti-commutation relations: ˙ ˙ Xα,α˙ , Xβ,β = ǫαβǫα˙ β (10.2) n o where α, α,˙ .. denote the spinor indices of the two SU(2) factors. Let 1 1 E = x2 Eα,α˙ = xXα,α˙ ∆= (xp + px) (10.3) 2 2 and take the generators of g0 to be of the form

α,β 1 α,α˙ β,β˙ β,β˙ α,α˙ M(1) = ǫα˙ β˙ X X + X X 4 (10.4) α,˙ β˙ 1 α,α˙ β,β˙ β,β˙ α,α˙ M(2) = ǫαβ X X + X X 4 They satisfy commutation relations of su(2) su(2) ⊕ α,β λ,µ λβ α,µ µα β,λ M(1) , M(1) = ǫ M(1) + ǫ M(1) h i α,˙ β˙ λ,˙ µ˙ λ˙ β˙ α,˙ µ˙ µ˙ α˙ β,˙ λ˙ M(2) , M(2) = ǫ M(2) + ǫ M(2) (10.5) h i α,β λ,˙ µ˙ M(1) , M(2) = 0 h i Their quadratic Casimirs are

˙ ˙ = ǫ ǫ M αλM βµ = ǫ ǫ M α˙ λM βµ˙ (10.6) I4 αβ λµ (1) (1) J4 α˙ β˙ λ˙ µ˙ (2) (2) Their sum is a c-number + = 3 . We use just one to construct I4 J4 − 2 generator of g+2 1 σ 3 9 F = p2 + + + σ (10.7) 2 x2 I4 4 8 and F αα˙ = i Eαα˙ , F (10.8) − Then ˙ ˙ F αα˙ , F ββ = 2ǫαβǫα˙ βF F αα˙ , F = 0 (10.9) h i Also

αα˙ ββ˙ αβ α˙ β˙ αβ α˙ β˙ α˙ β˙ αβ F , E = ǫ ǫ ∆ (1 3σ) iǫ M(2) (1 + 3σ) iǫ M(1) (10.10) h i − − − The parameter σ is left undetermined by the Jacobi identities. For σ = 0 the superalgebra D (2, 1,σ) is isomorphic to OSp(4 2, R) and for the values | σ = 1 it reduces to ± 3 SU (2 1, 1) SU (2) | ×

31 Acknowledgement: One of us (M.G.) would like to thank the organiz- ers of the workshop on “Mathematical Structures in String Theory” at KITP where part of this work was done. We would like to thank Andy Neitzke, Boris Pioline, David Vogan and Andrew Waldron for stimulating discussions and correspondence. This work was supported in part by the National Sci- ence Foundation under grant number PHY-0245337. Any opinions, ﬁndings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reﬂect the views of the National Science Foundation. We would also like to thank the referee for several suggestions for improving the presentation of the results.

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